Assumptions in dark-energy density parameter measurement

[center o bass]

I've understood that the main evidence for a non-zero $\Omega_\Lambda$ comes from supernova 1a measurements where one measures the redshifts along with the luminosity distances (equivalently magnitude) of the supernovae and, plots them against each other, then compares the result with theoretically derived curves for different values of cosmological parameters and find the parameter choices corresponding to the best fit. (One example of such plots are shown here.)

From what I have read, the only two parameters that are being varied is $\Omega_m$ and $\Omega_\Lambda$ from which one gets the result approximately $\Omega_\Lambda \approx 0.7$ and $\Omega_m \approx 0.3$ supporting the claim that the universe is flat since $\Omega_\Lambda + \Omega_m \approx 1$.

But here is the thing: since $\Omega_k$ was not varied, did we not already assume $\Omega_k = 0$ in the first place? Might it not have been the case that by also varying $\Omega_k$ this could've lead to a better fit with other parameter values than in the above result?

Or is there a good argument for why one can neglect $\Omega_k$ in the curve-fitting procedure?

 

[PeterDonis]

is there a good argument for why one can neglect $\Omega_k$ in the curve-fitting procedure?

As I understand it, it's because even if $\Omega_k$ isn't exactly zero, it's so close to zero (based on measurement that it can't significantly affect the analysis. The fact that the analysis ends up with $\Omega_{\Lambda } + \Omega_m = 1$ serves as a sanity check on the analysis; if that sum came out significantly different from 1, that would indicate a problem, since the analysis would be inconsistent with other measurements that indicate that the universe is extremely close to being spatially flat.

 

[Chalnoth]

I've understood that the main evidence for a non-zero $\Omega_\Lambda$ comes from supernova 1a measurements where one measures the redshifts along with the luminosity distances (equivalently magnitude) of the supernovae and, plots them against each other, then compares the result with theoretically derived curves for different values of cosmological parameters and find the parameter choices corresponding to the best fit. (One example of such plots are shown here.)

From what I have read, the only two parameters that are being varied is $\Omega_m$ and $\Omega_\Lambda$ from which one gets the result approximately $\Omega_\Lambda \approx 0.7$ and $\Omega_m \approx 0.3$ supporting the claim that the universe is flat since $\Omega_\Lambda + \Omega_m \approx 1$.

But here is the thing: since $\Omega_k$ was not varied, did we not already assume $\Omega_k = 0$ in the first place? Might it not have been the case that by also varying $\Omega_k$ this could've lead to a better fit with other parameter values than in the above result?

Or is there a good argument for why one can neglect $\Omega_k$ in the curve-fitting procedure?

Supernova measurements, especially early ones, didn't actually constrain \Omega_k very well. You could certainly let it vary, but the error bars on it are huge.

The way to resolve the discrepancy is to combine supernova data with other data, such as CMB data: the CMB constrains the curvature to be very close to flat. The basic picture here is that combining the CMB with nearby supernovae gives you a very long lever arm with which to measure curvature. To see why the long lever arm helps, consider the Earth: it's difficult to notice the curvature of the surface of the Earth while standing on the ground. But get far enough away, such as in low Earth orbit, and the curvature becomes quite apparent.

 

[Chalnoth]

By the way, this is illustrated by this plot of the various errors of some of the different measurements:
http://www.astro.virginia.edu/class/whittle/astr553/Topic01/t1_cos_combined.gif

The clusters are nearly-vertical because measurements of galaxy clusters mostly provide an estimate of the matter density, but don't give much information about the cosmological constant. The CMB's errors run very close to the "Flat" line because the CMB mostly gives information about the geometry of the universe, but very little information about the total density. The supernova data runs at nearly right angles to the CMB data because the supernova data hardly constrains curvature at all, but does give a tight constraint to the ratio between matter and dark energy.